From Extremal Combinatorics to Algorithms and Back

This project bridges two foundational fields in computer science and mathematics: Theory of Algorithms and Extremal Combinatorics. Algorithms are the cornerstone of modern computing, providing precise, step-by-step methods to solve complex problems efficiently. Extremal Combinatorics, a classical area of pure mathematics, investigates the boundaries of what is achievable within combinatorial structures, analyzing how large or small specific properties can be under given constraints. By integrating these areas, the project seeks to develop innovative computational techniques and theoretical insights that extend the frontiers of both fields. It focuses on leveraging recent breakthroughs that combine algorithmic approaches with classical results in extremal combinatorics. The anticipated outcomes are twofold: advancing state-of-the-art parameterized and approximation algorithms to address challenging computational problems with high efficiency, and formulating new stability theorems in extremal combinatorics, driven by practical algorithmic applications. By uniting these classical domains, the project not only enhances our understanding of foundational mathematical principles but also fosters innovation in algorithm design, paving the way for solutions to a wide range of algorithmic challenges.

The project lies at the intersection of two classical areas of Theoretical Computer Science and Mathematics: Theory of Algorithms and Extremal Combinatorics. Theory of Algorithms is a branch of computer science that focuses on designing, analyzing, and optimizing algorithms, which are step-by-step procedures or rules for solving computational problems efficiently. On the other hand, Extremal Combinatorics is a field of mathematics that studies the maximum or minimum values of invariants of a combinatorial objects satisfying certain constraints. By combining these two areas, the project aims to explore new computational methods and insights that could advance algorithmic theory and extremal combinatorics. Specifically, the project seeks to advance recent breakthroughs on algorithmic extensions of classical theorems in Extremal Combinatorics. The expected impact is twofold: advancing parameterized and approximation algorithms to new levels and developing new stability theorems in extremal combinatorics driven by algorithmic applications.